# Sylvester's Problem and Mock Heegner Points

@article{Dasgupta2017SylvestersPA, title={Sylvester's Problem and Mock Heegner Points}, author={Samit Dasgupta and John Voight}, journal={arXiv: Number Theory}, year={2017} }

We prove that if $p \equiv 4,7 \pmod{9}$ is prime and $3$ is not a cube modulo $p$, then both of the equations $x^3+y^3=p$ and $x^3+y^3=p^2$ have a solution with $x,y \in \mathbb{Q}$.

## 9 Citations

### Cube sums of the forms 3p and $$3p^2$$

- MathematicsMathematische Zeitschrift
- 2021

Let $$p\equiv 2,5\ \mathrm {mod}\ 9$$
be a prime. In this paper, we prove that at least one of 3p and $$3p^2$$
is a cube sum by constructing certain nontrivial Heegner points. We also establish the…

### Cube sums of form $3p$ and $3p^2$

- Mathematics
- 2018

Let $p\equiv 2,5\mod 9$ be an odd prime. In this paper, we prove that at least one of $3p$ and $3p^2$ is a cube sum by constructing certain nontrivial Heegner points. We also establish the explicit…

### On the $8$ case of the Sylvester conjecture

- MathematicsTransactions of the American Mathematical Society
- 2021

Let $p\equiv 8\mod 9$ be a prime. In this paper we give a sufficient condition such that at least one of $p$ and $p^2$ is the sum of two rational cubes. This is the first general result on the $8$…

### Integers expressible as the sum of two rational cubes

- Mathematics
- 2022

We prove that a positive proportion of integers are expressible as the sum of two rational cubes, and a positive proportion are not so expressible. More generally, we prove that a positive proportion…

### An explicit Gross–Zagier formula related to the Sylvester conjecture

- Computer ScienceTransactions of the American Mathematical Society
- 2019

This paper proves the <inline-formula content-type="math/mathml"> is not a cube and that a rational prime number such that it be a rationalprime number such as 4,7, 7, 9.

### N T ] 3 0 N ov 2 02 2 CYCLIC CUBIC EXTENSIONS OF Q

- Mathematics
- 2022

We determine the irreducible trinomials X − aX+ b for integers a, b which generate precisely all possible Galois extensions of degree 3 over Q. The proof, although involved, is elementary and one can…

### On the rank of cubic and quartic twists of elliptic curves by primes

- MathematicsJournal of Number Theory
- 2022

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This paper proves the <inline-formula content-type="math/mathml"> is not a cube and that a rational prime number such that it be a rationalprime number such as 4,7, 7, 9.